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The question provided involves proving that triangles \( \Delta RUV \) and \( \Delta TSV \) are congruent. Let's break down the proof and complete it.

### Given Information:
- \( V \) is the midpoint of \( RT \).
- \( RT = RU \) (Given).
- \( ST = SV \) (Given).
- All right angles are congruent (Given).
- \( \angle RUV \cong \angle TSV \) (Vertical Angle Theorem).

### Proof:

We have already deduced that \( \Delta RUV \cong \Delta TSV \), and the table of statements and reasons is almost complete. However, the final reasoning is missing.

The triangles share a vertical angle at \( V \), and we know that \( V \) is the midpoint, so \( RV = VT \), making the two sides congruent. Based on this, we can conclude that the two triangles are congruent by **SAS (Side-Angle-Side)** congruence theorem, since:

1. \( RV = VT \) (by midpoint definition),
2. \( \angle RUV \cong \angle TSV \) (by vertical angles),
3. \( RU = ST \) (Given).

### The correct answer is: **SAS**.

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Do you want more details or have any other questions?

Here are 5 related questions to expand on this concept:
1. How does the SAS postulate prove triangle congruence?
2. What is the difference between the SAS and ASA postulates?
3. Could you use a different method (such as ASA) to prove the triangles congruent?
4. Why do vertical angles always result in congruent angles?
5. How does the midpoint definition help in proving congruence in triangles?

**Tip:** When using congruence theorems like SAS, always check for matching sides and angles between the triangles to ensure that they are properly aligned for the proof.

V is the midpoint of RT. Complete the proof that ΔRUV ≅ ΔTSV.

Learn how to prove that triangles ΔRUV and ΔTSV are congruent using the Vertical Angle Theorem and Midpoint Theorem. This problem is suitable for students in Grades 8-10 and helps understand triangle congruence through SAS (Side-Angle-Side) postulate.

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